Values of the cosine integral $\operatorname{Ci}(x)$

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Numbers

$x$

$\operatorname{Ci}(x)$

$1\pi/2$:

0.4720006514395686507776061076141278365073305430183618818683837189938580377695313085093379707604929212

comment: The argument is specified by (4).

$3\pi/2$:

-0.1984075606923580425064010681429755266686852454060932732213938455273927624051632573606121330930035448

comment: The argument is specified by (4).

$5\pi/2$:

0.1237722754032595696107486910633392875005020097982834974585363670985119061147564893812595388685801941

comment: The argument is specified by (4).

$7\pi/2$:

-0.08956402592847867349131953509002327428896068115645728739656862463034679706938623286201020448106483069

comment: The argument is specified by (4).

$9\pi/2$:

0.07006507730931906630347120533941851557766139280656013176994117778050315599629877344678089202423551018

comment: The argument is specified by (4).

$11\pi/2$:

-0.05750107539948451110050976360680657494368785938598377996030275068453822632540675812996084559272919795

comment: The argument is specified by (4).

$13\pi/2$:

0.04874220212933136246948333524061119486210574361212554848665823066320255046044103484557962501459745386

comment: The argument is specified by (4).

$15\pi/2$:

-0.04229156391059048648235693916706125016059727306081014642951580372499231291876911799698229710875155177

comment: The argument is specified by (4).

$17\pi/2$:

0.03734488781135967601463250051546232007931392660851124334064130875761485475711213276459468977063501744

comment: The argument is specified by (4).

$19\pi/2$:

-0.03343205197080954446304702156141163926601186665938765259496247947281544393750576187309528152422926444

comment: The argument is specified by (4).

$21\pi/2$:

0.03026010543586373637590429468342588929133621741350165635583090545472572015233243116209453542978569639

comment: The argument is specified by (4).

$23\pi/2$:

-0.02763709003883375302610623016855633624104183839071178276332613962493146079746058647215914662942877581

comment: The argument is specified by (4).

$25\pi/2$:

0.02543201748324884596480396165588348694308700222698842532332629440813593913881612135522000934206111800

comment: The argument is specified by (4).

$27\pi/2$:

-0.02355246539524913385618601625010607276749544348980968386451104600193258604179595482714824174925207424

comment: The argument is specified by (4).

$29\pi/2$:

0.02193136848997190751180563562175639889183116860925226208072068054777360443874992989767524933038348343

comment: The argument is specified by (4).

$31\pi/2$:

-0.02051888677441717489616850547541487208943252787554423542003246815402502742163959851068401193505183477

comment: The argument is specified by (4).

$33\pi/2$:

0.01927721253541326499776219586843088604185939209806317423591130851245153160670321656802483833026992506

comment: The argument is specified by (4).

$35\pi/2$:

-0.01817714811239993945669094200407734626030405439914606559085623227143080295097198563744178919896791402

comment: The argument is specified by (4).

$37\pi/2$:

0.01719578822569244988110969658920038310552350794301186720778157394096570104539928926555960082009527008

comment: The argument is specified by (4).

$39\pi/2$:

-0.01631491235890297963143481730785287243197920856323752651290328374698678038776512611224804527487775828

comment: The argument is specified by (4).

1/4:

-0.8246630625809456530858650345190850733960949224566059347682517446693093913051280769434625217991471300

comment: A near-origin or moderate-argument reference; see (5).

1/2:

-0.1777840788066129013358102710705690780905194748126219686668253575951272650159020967709876446027217324

comment: A near-origin or moderate-argument reference; see (5).

0.3374039229009681346626462038891507699975780325857318948013185424361303300250560528968481830973229946

comment: A near-origin or moderate-argument reference; see (5).

0.4229808287748649956985651531982558941357377563061876881101624557656023036996416214224698023378797897

comment: A near-origin or moderate-argument reference; see (5).

-0.1409816978869304116391448986940359267129683554616938615858568244712435627860090592938105570169452208

comment: A near-origin or moderate-argument reference; see (5).

0.1224338825320095572922959582687350270353737871758982206298723514276775852453312174113305622412029130

comment: A near-origin or moderate-argument reference; see (5).

$1\pi$:

0.07366791204642548599010096523014967186987746232861805026595503406923175843143057138365844278326088243

comment: A complementary oscillation-phase reference; see (6).

$2\pi$:

-0.02256066174634606764353877854304643364737004780062353257256679497818099416959119913784964043910454767

comment: A complementary oscillation-phase reference; see (6).

$3\pi$:

0.01062020296525028813845526383035790572060871949723742441958482187516244842233688065836528865674315511

comment: A complementary oscillation-phase reference; see (6).

$4\pi$:

-0.006116639131919572887954084778690916002579865093634802435451478951713348537531691449122519643663667757

comment: A complementary oscillation-phase reference; see (6).

Definition

The cosine integral is $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos t-1)/t\,\mathrm{d}t$ [4]. This table gives its values at intrinsic and bounded reference arguments.

Parameters

$x$

— argument ($x>0$)

Formulas

(1)

For $x>0$, $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$ and $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$ [4].

(2)

The functions $\operatorname{Si}$ and $\operatorname{Shi}$ are odd [4]. With the principal branch of the logarithm, for $x>0$, $\operatorname{Ci}(-x)=\operatorname{Ci}(x)+i\pi$ and $\operatorname{Chi}(-x)=\operatorname{Chi}(x)+i\pi$ [5].

Comments

(3)

The constant $\gamma$ in the definition is the Euler-Mascheroni constant.

(4)

Differentiation gives $\operatorname{Ci}^{\prime}(x)=\cos(x)/x$ [8]; hence its positive extrema occur at $x=(2n-1)\pi/2$, $n\geq1$.

(5)

The exploratory reference set $x=1/4,1/2,1,2,4,8$ provides sparse positive inputs near the origin and into the oscillatory regime. It complements the extrema without claiming a special use for every input. The cosine integral uses its real branch on $x>0$ [9].

(6)

The additional arguments $n\pi$, $1\leq n\leq4$, sample the complementary phase positions between successive stationary points. They are selected from the phase of the defining sine or cosine integrand, with a short noncanonical cutoff, rather than from a decimal mesh [9].

(7)

The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.

Programs

(P1)

Python

from generate import generator
from numberdb._write import to_text

for params, result in generator.all_entries():
    print(params["x"], to_text(result["number"], generator.digits_for(params)))

Links

[1]

Wikipedia: Exponential integral

[2]

Wikipedia: Logarithmic integral function

[3]

Wikipedia: Trigonometric integral

[4]

DLMF 6.2: Definitions and Basic Properties

[5]

DLMF 6.4: Analytic Continuation

[6]

DLMF 8.19: Generalized Exponential Integral

[7]

OEIS A099282: decimal expansion of $\operatorname{Ci}(1)$

[8]

DLMF: definitions and natural argument families

[9]

DLMF 6.2(ii): sine and cosine integrals

Similar tables

Values of the sine integral $\operatorname{Si}(x)$ — the companion trigonometric integral

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — $\operatorname{Chi}$ is the same integral with $\cosh$ in place of $\cos$

Wilbraham-Gibbs constant — stores $\operatorname{Si}(\pi)$ and its normalisations

Stieltjes constants — contains $\gamma_0$, the Euler-Mascheroni constant appearing in $\operatorname{Ei}$, $\operatorname{Ci}$ and $\operatorname{Chi}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the twenty extremal heights at $x=(2n-1)\pi/2$, $1\leq n\leq20$; the selection is not exhaustive over the real domain. In addition, it includes $x=1/4,1/2,1,2,4,8$ and the complementary phase arguments $n\pi$ for $1\leq n\leq4$. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.)

How they were obtained:

All retained values and metadata are preserved, including their precision. New nonexact values have 100 significant digits. balanced_values.py reproduces the complete selection in balanced-selection.json with Arb balls, including uncertainty in exact irrational arguments and certified roots. A 640-bit guard is supplemented by measured accuracy checks and precision escalation.

Every written interval contains a freshly recomputed certified enclosure; every value is also checked with independent mpmath computations at 50 and 100 decimal guard digits. Exact origin anchors stay exact. Structural identities, root endpoint signs where applicable, and selected transformed definite integrals supplement these checks. See balanced-checks-evidence.json. Earlier checks-evidence.json and source attachments describe earlier selections and remain for provenance.